I wouldn't really consider 12, 13, and 14 to be exclusions since, as the "Tips and notes" (https://www.duolingo.com/skill/ru/Time-and-Numbers) point out, they are numbers larger than four and therefore take the genitive plural.
Look at how they are formed in Russian:
- 22 is два́дцать две́ [feminine] or два́дцать два [masculine or neuter] (therefore, it uses the noun form for две/два),
- 23 is два́дцать три (therefore, it uses the noun form for три),
- 42 is два́дцать два/две,
- 53 is пятьдеся́т три,
- 64 is шестьдеся́т четы́ре.
At the same time, numbers from 11 to 19 are formed differently (the pattern was like 'one-on-ten', 'two-on-ten'):
- двена́дцать ends in -дцать, which is a contracted form of десять and therefore you use the noun form for десять,
- трина́дцать ends in -дцать,
- четы́рнадцать ends in -дцать.
The way these numbers are formed is really an exception (for ten, you use 'one-on-ten' pattern; for other tens, you use 'twenty one' pattern), but the noun form choice doesn't seem too illogical to me.
Because it follows the hard consonant (and is not separated by a pause). «И» cannot be pronounced after a hard consonant, so when it follows a hard consonant, it becomes ы.
When you add a prefix to a word beginning with И, it becomes Ы: без 'without' + изве́стный 'famous' = безызве́стный 'obscure'.
However, when it's written as a separate word, we don't change the spelling. However, if separate words are pronounced without a pause, we still pronounce [ы]: без 'without' + изве́стий 'news' = без [ы]зве́стий 'without news'.